We study the decay of correlation between locally constrained independent random variables in the local lemma regimes. the distribution defined by constraint satisfaction problems (CSPs) in the local lemma regime. For atomically constrained independent random variables of sufficiently large domains, we show that a decay of correlation property holds up to the local lemma condition $pD^{2+o(1)}\lesssim 1$, asymptotically matching the sampling threshold for constraint satisfaction solutions [BGG+19,GGW22]. This provides evidence for the conjectured $pD^2\lesssim 1$ threshold for the "sampling Lov\'{a}sz local lemma". We use a recursively-constructed coupling to bound the correlation decay. Our approach completely dispenses with the "freezing" paradigm originated from Beck [Bec91], which was commonly used to deal with the non-self-reducibility of the local lemma regimes, and hence can bypass the current technical barriers due to the use of $\{2,3\}$-trees.

翻译：我们研究了局部引理区域中局部约束独立随机变量之间的相关性衰减。在局部引理条件下，由约束满足问题（CSPs）定义的概率分布中，对于具有足够大值域的原子约束独立随机变量，我们证明了当局部引理条件$pD^{2+o(1)}\lesssim 1$成立时，相关性衰减性质始终存在，这一条件在渐近意义上匹配了约束满足问题解的采样阈值[BGG+19, GGW22]。这为"采样Lovász局部引理"中推测的阈值$pD^2\lesssim 1$提供了证据。我们采用递归构造的耦合方法来界定相关性衰减。该方法完全摒弃了Beck [Bec91]提出的"冻结"范式（该范式常用于处理局部引理区域中的非自归约性问题），从而能够规避当前因使用$\{2,3\}$-树而面临的技术瓶颈。